S ∞ =
∞
∑ = a * r ^ n = a ⁄ ( 1 – r )
n=1
In this case a = 2 , infinite sum = 10 so:
S ∞ = 10 = a ⁄ ( 1 – r )
10 = 2 ⁄ ( 1 – r ) Multiply both sides by 1 - r
10 * ( 1 – r ) = 2
10 * 1 – 10 * r = 2
10 - 10 r = 2 Subtract 10 to both sides
10 - 10 r - 10 = 2 - 10
- 10 r = - 8 Divide both sides by - 10
- 10 r / - 10 = - 8 / - 10
r = 0.8
The n-th term of a geometric series with initial value a and common ratio r is given by:
an = a * r ^ ( n − 1 )
a1 = a * r ^ ( n − 1 ) = 2 * 0.8 ^ ( 1 − 1 ) = 2 * 0.8 ^ 0 = 2 * 1 = 2
a2 = a * r ^ ( n − 1 ) = 2 * 0.8 ^ ( 2 − 1 ) = 2 * 0.8 = 1.6
a3 = a * r ^ ( n − 1 ) = 2 * 0.8 ^ ( 3 − 1 ) = 2 * 0.8 ^ 2 = 2 * 0.64 = 1.28
a4 = a * r ^ ( n − 1 ) = 2 * 0.8 ^ ( 4 − 1 ) = 2 * 0.8 ^ 3 = 2 * 0.512 = 1.024
Describe an infinite geometric series with a beginning value of 2 that converges to 10. What are the first 4 terms of the series?
Last question and my mind is fried, can someone please help me? Thanks
1 answer
2 answers